On the structure of boolean functions with small spectral norm

  • Authors:

  • Amir Shpilka;Avishay Tal;Ben lee Volk

  • Affiliations:

  • Technion - Israel Institute of Technology, Haifa, Israel;Weizmann Institute of Science, Rehovot, Israel;Technion - Israel Institute of Technology, Haifa, Israel

  • Venue:
  • Proceedings of the 5th conference on Innovations in theoretical computer science
  • Year:
  • 2014

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Abstract

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of ƒ is ||ƒ||1 = ∑α|ƒ(α)|). Specifically, we prove the following results for functions ƒ :{0, 1}n → [0, 1}with ||ƒ||1 = A. There is a subspace V of co-dimension at most A2 such that ƒ|v is constant. ƒ can be computed by a parity decision tree of size 2A2n2a. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) ƒ can be computed by a De Morgan formula of size O(2A2 n2A+2) and by a De Morgan formula of depth O( A2 + log(n) • A). If in addition ƒ has at most s nonzero Fourier coefficients, then ƒ can be computed by a parity decision tree of depth A2log s. For every ε 0 there is a parity decision tree of depth O(A2 + log(1/ε)) and size 2O(A2)} • min{ 1/ε2,O(log(1/ε))2A} that ε-approximates ƒ. Furthermore, this tree can be learned, with probability 1--δ, using poly(n, exp(A2), 1/ε,log(1/δ)) membership queries. All the results above (except ref{abs:DeMorgan}) also hold (with a slight change in parameters) for functions f : Znp → {0, 1}.